Gibbard's theorem

  • in the fields of mechanism design and social choice theory, gibbard's theorem is a result proven by philosopher allan gibbard in 1973.[1] it states that for any deterministic process of collective decision, at least one of the following three properties must hold:

    1. the process is dictatorial, i.e. there exists a distinguished agent who can impose the outcome;
    2. the process limits the possible outcomes to two options only;
    3. the process encourages agents to think strategically: once an agent has identified their preferences, they have no action at their disposal that would best defend their opinions in any situation.

    a corollary of this theorem is gibbard–satterthwaite theorem about voting rules. the main difference between the two is that gibbard–satterthwaite theorem is limited to ranked (ordinal) voting rules: a voter's action consists in giving a preference ranking over the available options. gibbard's theorem is more general and considers processes of collective decision that may not be ordinal: for example, voting systems where voters assign grades to candidates.

    gibbard's theorem is itself generalized by gibbard's 1978 theorem and hylland's theorem, which extend these results to non-deterministic processes, i.e. where the outcome may not only depend on the agents' actions but may also involve a part of chance.

  • overview
  • formal statement
  • examples
  • notes and references
  • see also

In the fields of mechanism design and social choice theory, Gibbard's theorem is a result proven by philosopher Allan Gibbard in 1973.[1] It states that for any deterministic process of collective decision, at least one of the following three properties must hold:

  1. The process is dictatorial, i.e. there exists a distinguished agent who can impose the outcome;
  2. The process limits the possible outcomes to two options only;
  3. The process encourages agents to think strategically: once an agent has identified their preferences, they have no action at their disposal that would best defend their opinions in any situation.

A corollary of this theorem is Gibbard–Satterthwaite theorem about voting rules. The main difference between the two is that Gibbard–Satterthwaite theorem is limited to ranked (ordinal) voting rules: a voter's action consists in giving a preference ranking over the available options. Gibbard's theorem is more general and considers processes of collective decision that may not be ordinal: for example, voting systems where voters assign grades to candidates.

Gibbard's theorem is itself generalized by Gibbard's 1978 theorem and Hylland's theorem, which extend these results to non-deterministic processes, i.e. where the outcome may not only depend on the agents' actions but may also involve a part of chance.